Consider the signal to be piecewise cubic,
with smooth connections at the nodes.
It uses a cubic curve which has node values
x0 at 0 and x1 at 1 and derivatives
(x1-xm1)2 and (x2-x0)2, respectively.
You can see how it works
if you evaluate the expression for t=0 and t=1
as well as the derivative at these points.